The Fourier Transform 

Thank you!!

@@Eigensteve Seriously. I too wish my instructors had put in half as much effort in being concise and coherent as you have. I've been watching your videos in order for about a week now.

Glad you liked it!!

Glad you enjoy it!

Thanks!

Hallo! Which city?

You're welcome!

no, the Fourier transform of a period function will diverge. It can, however, be defined if you know about the delta 'function'.

He at the beginning took w_k = k(pi)/L, then he wrote it as kΔw. Where Δw= PI/L. In the limit of Δw -> 0, inside the integral, he easily replaced w_k from the first equation. You can also write it as w only in the continuous limit.

yes it had to be so!

Indeed!

The Riemann integral is simply defined by the limit of a Riemann Sum as the delta variable approaches zero. The instructor just recognized the form and replaced it with the integral notation. Not sure if that answers your question.

@@kylegreen5600 I don't understand. When can you integrate a Fourier series term by term?

@@NoNTr1v1aL Could you respond with a rough time in the video where you're not following the steps and I'll try to help.

@@NoNTr1v1aL Your question is different from what Dr. Brunton did: like Kyle said, he's just recognizing a Riemann sum that already exists, he's not integrating the Fourier series. It's just a definition: as L -> inf, the series turns into an integral. Your question is different and gets into deep waters. The short answer is: if f is "nice enough" (e.g. C^inf smooth.) The long answer: this is the entire subject of Fourier and harmonic analysis. In general, integrating and differentiating series term by term depends on the analytic properties of the f being expanded, the types of convergence (uniform, pointwise, etc.) and it only gets more subtle with Fourier. The Gibbs phenomenon is already an example of this: a jump discontinuity in f screws up uniform convergence of the fourier series, and if you remember, Dr. Brunton mentioned that the "top hat" function was related to the derivative of the "triangle hat" function (which was continuous, but not continuously differentiable), so already you can see some of these subtleties creeping in. Check out Stein and Sharkachi's books, Tao's books, etc.

when the integrable function is uniformly continuous, then we can do like this interchanging.

Same question here. The part of taking the Riemann integral makes sense on the expression as a whole, but I'm confused as to what justifies that substitution (hiding the fact that the delta omega has shrunk to zero). Probably has to do with the fact that the k is infinetly summed (and then transformed to Riemann integral).

Sorry to hear that, but don’t despair! A little more background in linear algebra and vectors would likely help. You could check out my courses to see where this fits in: faculty.washington.edu/sbrunton/me564/ and faculty.washington.edu/sbrunton/me565/

@@Eigensteve Thanks man!

bro wtf its just mirrored camera, of course he's writing in the right direction

@@alegian7934 jokes

@@Thespookygoat oof I wooshed rather hard there... sry this is math after all

??